Algebraically Skew Embeddings, Positivity of Vector Bundles, and Application of SheavesDay, Andy Kuangalgebraically skew embeddingTerracini locusalgebraic hyperbolicityprobability sheafMathematicsPHDDifferent areas in geometry and in mathematics in general, often encounter similar problems, yet approach these problems with tools of different strength, and obtain results of different flavors. On the one hand, the flexible structure of smooth manifolds often leads to surprising constructions, and thus make seemingly straightforward problem difficult. On the other hand, in the areas such as algebraic geometry and combinatorial topology, rich and rigid structures often lead to complexities of an entirely different flavor. Nevertheless, developments in one subject frequently inspire parallel theories in others. Such interactions can provide new approaches to longstanding problems or in some cases leads to new aspects of the area that would not otherwise be considered before. In this dissertation, we include three projects that lie in between differential topology, algebraic geometry, and combinatorial topology. The main project of this dissertation is the one of the algebraically skew embedding problem, which is inspired by the totally skew embedding problem in differential topology. An embedding of a smooth manifold in a Euclidean space is said to be totally skew if any pair of its embedded tangent spaces are disjoint and contain no parallel lines. The totally skew embedding problem seeks the minimum ambient dimension of the totally skew embedding for a given manifold. In Chapter 3 and 4 we study the intersection theoretic aspect of the Gauss map and the blowups of Grassmannians. Then we establish an algorithm for the lower bounds of the minimum ambient dimension of algebraically skew embedding of a given smooth complex variety, in terms of its Chern classes. The second project in this dissertation concerns the algebraic hyperbolicity of subvarieties in products of projective spaces. A complex variety $X$ is said to be algebraically hyperbolic if it admits a positive real number $\epsilon$ and an ample divisor $H$ such that for any curve $C\subset X$ we have $2g(C)-2>\epsilon H\cdot C$. The algebraic hyperbolicity is the algebraic analogue of the Kobayashi and Brody hyperbolicity in complex geometry. In Chapter 5 we generalize the result for hyperbolicity of high degree hypersurfaces to the zero locus of vector bundles of homogeneous varieties in products of projective varieties. The third project of the dissertation concerns the probability (pre)-sheaves on the power set of finite sets. In Chapter 6 we investigate the topological and combinatorial obstructions that stop the global sections of a probability (pre)-sheaf over finite sets being realized by global distributions.Pierre-Emmanuel Jabin, Program Head/ChairJack Huizenga, Major Field MemberSlava Rotkin, Outside Unit & Field MemberSergei Tabachnikov, Chair & Co-Dissertation AdvisrJohn Lesieutre, Dissertation Co-AdvisorVladimir Itskov, Major Field Memberopen_access2026-07-07T14:45:42Zhttps://etda.libraries.psu.edu/catalog/22431bpd5341